Anyonic order parameters for discrete gauge theories on the lattice

TL;DR

This paper introduces new gauge-invariant non-local order parameters for non-abelian discrete gauge theories on lattices, linking them to the excitation spectrum from the quantum double of the finite group, and demonstrates their importance in probing phase structures.

Contribution

It proposes a novel family of order parameters combining magnetic flux and electric charge, extending traditional Wilson and 't Hooft operators, for analyzing phase transitions in discrete lattice models.

Findings

Order parameters correspond to the excitation spectrum of D(H).

They are essential for Monte Carlo simulations of phase structures.

Reduce to Wilson and 't Hooft operators in special cases.

Abstract

We present a new family of gauge invariant non-local order parameters for (non-abelian) discrete gauge theories on a Euclidean lattice, which are in one-to-one correspondence with the excitation spectrum that follows from the representation theory of the quantum double D(H) of the finite group H. These combine magnetic flux-sector labeled by a conjugacy class with an electric representation of the centralizer subgroup that commutes with the flux. In particular cases like the trivial class for magnetic flux, or the trivial irrep for electric charge, these order parameters reduce to the familiar Wilson and the 't Hooft operators respectively. It is pointed out that these novel operators are crucial for probing the phase structure of a class of discrete lattice models we define, using Monte Carlo simulations.